FONDATĂ 1976 THE ANNALS OF “DUNAREA DE JOS” UNIVERSITY OF GALATI. FASCICLE IX. METALLURGY AND MATERIALS SCIENCE N0. 4 – 2010, ISSN 1453 – 083X

DIMENSIONLESS NUMBERS IN THE ANALYSIS OF HYDRODYNAMIC INSTABILITY OF INTERFACE STEEL - SLAG AT MICROSCALE IN REFINING PROCESSES

Petre Stelian NITA ”Dunarea de Jos” University of Galati email: pnita @ugal.ro

ABSTRACT

In evaluation of the local hydrodynamics of interface, in steel-slag refining processes influenced by solutal effects, adequate scales of length and time satisfying certain conditions are necessary to get a correct and suggestive image of the weight of factors possible to be used in industrial technologies. These could be obtained based on already established common dimensionless numbers but carefully applied in a specific manner, taking into consideration particularities of the established interface and the main involved process. Based of the general philosophy of building dimensionless artificial conglomerates, a new dimensionless group Ni = Ma ⋅ Bo is proposed in this paper as a consequence of needs to evaluate particularly the effects of capillarity actions at interface, especially of solutal origin, in presence of other physico-chemical and physical actions, especially when the effects of capillarity prevail over all others.

KEYWORDS: steel refining, hydrodynamic instability of interface, length scale, time scale, dimensionless numbers.

1. Introduction extend because of the final purpose of the applied treatments which is to provide a certain appreciable Despite the fact that capillarity is recognized as high amount of refined steel, presenting a high level an important phenomenon in hydrodynamics of of purity concerning harmful elements. Oxygen, interface, only extremely rare and more than prudent sulphur and phosphorus are the most frequently target approaches of this subject are present in scientific and elements of treatments applied to steels using slags. technical papers dealing with process at interfaces in Several problems in analyzing the influence exerted metallurgical systems steel-slag. Therefore in the by the presence of surface active solute in the liquid following paragraphs there is a short introduction in phases composing the refining system steel-slag are the specific problems and aspects of this class of analyzed in a simplified manner due to their low systems and this represents an attempt of the author to specific concentrations [1][2]. open in a certain measure new doors in the At low levels of concentrations of surface active fundamentals of some new metallurgical processes of solutes, their influence on density is insignificant, also refining based on the advanced knowledge. As a main the corresponding aspects of buoyancy due to approach, specific dimensionless numbers are variations of density. Up to values of maximal presented and comments are made on the connections concentrations of 2 mass% sulphur in slags and 0.02% with the hydrodynamics of interface steel-slag. mass% sulphur in steel, which are specific to Solutocapillarity plays an important and desulphurization process at industrial scale, there are particular role in the hydrodynamics of interface not reported sensible variations of density in the between liquid phases of technological interest in respective liquid phases, compared to the densities of steel refining under slags. the same liquids in absence of sulphur. Therefore it is The amplitude of the local fluctuations of possible that aspects of buoyancy could be totally concentrations at interface between steel and slag is neglected. limited compared to the possible local fluctuations of The solutocapillarity due to surface tension temperature in other systems or when strong gradients gradient produced by perturbation of concentration is of temperature are imposed externally by special a factor of instability while the surface tension acts to techniques or devices. Supplementary, the depths of stabilize the interface; viscosity, diffusion and the layers of steel and slag are of a certain established gravitation are also stabilizing actions. Viscous

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diffusion acts to dampen the concentration decomposed into normal components (normal stress fluctuations and the associated fluid flow. The gravity and pressure) and a tangential component (shear acts physically to flatten the interface and thus to stress). Boundary conditions at interface lead to the stabilize the deformational perturbations. The following physical considerations [3]: presence of a heavier liquid phase below a lighter one 1. jump in fluid pressure is balanced by interface (as it is the case of steel under slag) is also a tension; stabilizing factor. 2. jump in fluid stress is balanced by surface If it is intended to put in evidence the absolute gradient of interface tension (gradient along the role of the destabilizing factor–solutocapillarity in interface) ∇Sσi . promoting flow, it is necessary that other destabilizing Macroscopic interfacial no-slip conditions break factors or factors damping the solutocapillarity effects when at microscale, processes due to surface tension to be minimized at certain level, bellow a certain gradient, become relevant as magnitude. Analysis of threshold value of the involved parameter. This could this class of processes could be performed in be performed selecting or imposing values of different conditions of the existence of a deformable interface parameters in a physically acceptable range and but which doesn’t deform because of the small weight leading finally to an adequate time scale. Several of factors leading to deformations. This situation general remarks regarding properly named operations could be encountered at least in the incipient state of in scaling and in its preliminary stages and considered flow, or during time intervals where considerations are useful by their close relation with required conditions are accomplished. This is the case the technological and physical reality and utility. In when physical actions producing deformations are not this sense, in an such analysis principle of in acts a dominant in comparison to others. It results that, in certain incertitude; imposing too restrictive conditions conditions of adequate scaling of length and time of to obtain favorable conditions of prevalence of a different actions, computations of parameters certain physical action, compared to another, not only regarding the resulting particular and specific flow are that this could not serve too much, but it could block realistic and possible to be made. the utility of the respective scale. For example, the The known dimensionless numbers are limited as characteristic time scale which is dependent upon the possibilities to give a good image of the physical characteristic length at different powers. These reality in the problems mentioned in title. Several of negative aspects could be totally avoided by a correct them use a characteristic length with a certain scaling operation based on a correct meaning of the specificity which differs from a phenomenon to processes and on an adequate selection of the another (it is well known that there are different dimensionless numbers used in these cases. Further independent scales of length). In fact, the main going there is also the possibility to introduce a new problem in the analysis of local instability produced dimensionless number taking into account the need to by different actions, including the mass transfer of a evaluate the simultaneous complex action of many surface active element, consists in a correct and several forces involved in interfacial flow and specific scaling of length, velocity and time. It is dynamics as it is more and more encountered in known that every physical or physicochemical action literature. is characterized by scales of length, time and velocity which are almost independent among them, but when 2. Dimensionless numbers involved in the actions are simultaneous, the fastest action has the interfacial hydrodynamic instability largest probability to occur. If it is analyzed the occurrence of a certain physical or a physicochemical 2.1 Principles and particularities action it is necessary to scale the time and the Technological systems in steel refining under conditions when it is the dominant action with a slags could be included in the class of complex certain high degree of certitude. This is usually made process of multi-phase flow, where a fluid interface establishing the condition when the time scale is separates immiscible phases, presenting a certain shorter with at lest an order of magnitude, compared surface tension(σ ,σ ) and leading to certain to other concurrent actions. From this condition a steel slag characteristic length scale results, and in certain interface tension(σ i ).The fluid interface is conditions, also a velocity scale results. Other times it deformable, it has an unlimited extend, reported to the is possible to start the scaling from velocity, instead dimensions of the interfacial layer thickness and it is of time and then the characteristic length and time unbounded at wall, that means the interface is in the result on this base. free-slip condition. This is generally valid if the This kind of analysis is often the single way to get friction at interface could be neglected or included in some information on the possible behavior of a other quantified physical factors. Therefore, there is a system and to perform a simulation concerning the force balance at slag-steel interface and this could be behavior of the considered system. In the system

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steel-slag at liquid phases temperature it is obvious ρ ⋅ a ⋅ d 2 the difficulty of any direct observation. Some enough Bo = (2) σ valuable data could be obtained using the rapid Where: ρ − is the density, or the density solidification method of samples containing portions of interface steel-slag. difference between fluids; a − is the acceleration of Many dimensionless numbers represent ratios of the body force. two physical actions (forces) or of two families of Mainly Bo dimensionless number could be actions. This level of synthetic representations despite derived from the general dimensionless number N extremely useful, does not satisfy the actual introducing gravitation scaling of the pressure P. necessities, mainly consisting in the enlargement of It quantifies the importance of body forces over the set of actions evaluated in such manner. the surface tension forces, connected with the role of At small time and length scales, different equilibrium capillarity (σ ). When the acceleration a phenomena leading to thermodynamic equilibrium is gravity acceleration g, The Bo number is the most exhibit strong particularities affecting the common comparison of gravity and surface tension hydrodynamics in a dramatic manner. Local effects, usually used to find the characteristic length fluctuations of temperature and concentration at the scale in complex scaling problems.: interface or at free open surface alter locally the ρ ⋅ g ⋅ d 2 surface tension and if these local inhomogeneities are Bo = (3) continual and persistent enough time, they could give σ a spontaneous flow (Marangoni effect). Therefore it is Surface tension dominates the flow at small necessary always to relate the analyzed processes to values of d , this being the single parameter of a given adequate scales of time and of length. In such system where the density, and gravity acceleration are conditions the surface tension gradients constant and there is a certain variability of surface tension upon the concentration of a surface active ∂σ ∂x contribute to a force proportional to the solute. At interface steel-slag the competition between thermal coefficient ∂σ ∂T or to the concentration stabilizing factors of interface is from far in favor of coefficient ∂σ ∂c of interface tension. In this paper surface tension if the length scale is small enough. A value Bo<1 indicates that surface tension dominates. only the influence of concentration will be The problem which is still persistent is to establish the considered, based on the physical reality consisting in length scale d. At Bo=1 the capillary length scale is much more values of the thermal diffusion and obtained: conductivity in slag and steels, compared to their 1 2 mass equivalents. Lc = (σ ρ ⋅ g) (4) At an interface between two immiscible 2.2. Dimensionless numbers derived from superposed liquids of density the normal component of the free slip ρtop = ρ1 and ρbottom = ρ2 the capillary length is boundary condition obtained replacing ρ by Δρ = ρ2 − ρ1: The normal component of the free-slip boundary L = (σ Δρ ⋅ g)1 2 (5) conditions gives by scaling the Laplace number(La); c it represents a ratio of surface tension and inertia If the interface tension gradient is persistent forces to the viscous forces, expressing the enough time and is continual on interface, it could be momentum-transport (especially dissipation) inside a taken into account also in the relation giving the fluid. La dimensionless number is linked to the free capillary action of the interface tension gradient of convection inside of immiscible fluids and is defined solutal origin: 1 2 by the relation: Lc = (Δσ Δρ ⋅ g) (6) σ ⋅ ρ ⋅ L σ ⋅ L At lengths L ≤ L the capillarity prevails over La = = (1) c μ 2 ρ ⋅ν 2 gravity. This limit extend of action of capillarity must where:σ − is surface tension; ρ − is be corrected, taking into consideration the values where the capillarity prevails over other competing density; L − is characteristic length; μ − is dynamic actions. viscosity;ν − is kinematic viscosity, also called Other forms expressing the ratio between gravity momentum diffusivity. forces and surface tension are also used in the form of Another form is Ohnesorge number Oh=La-2. other dimensionless numbers. Bond dimensionless number (Bo), also called Influences exerted by the solutal effects, reflected the static Bo number in opposition with another form in the interfacial tension coefficient, using the called dynamic Bond dimensionless numb er is given Rayleigh number and the dynamic Bond (Bod) by the following ratio [5]: number, where the relative importance of buoyancy is

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evaluated, are difficult to be put into evidence. replacing V 2 = g ⋅ d 2 , the dimensionless number We Despite the fact that the Rayleigh dimensionless number usually refers to thermal effects on buoyancy can be written as: Δρ ⋅ g ⋅ d 2 due to the modifications of concentration, it can also We = (13) refer to the solutal effects but only related to the σ modifications of the density due to this kind of The characteristic length in the case of a drop is effects. its diameter. When inertial effects dominate over viscous effects, the stress could be expressed in Capillary dimensionless number (Ca), also dimensionless form based on the inertial scale.We called as crispation dimensionless number (C) [6] is number, which expresses the relative influence of specific to situations where the scale of stresses in inertia and hydrodynamic pressure related to the presence of surface tension is a viscous one. capillary pressure, it is particularly useful in analyzing It is given by the relation: fluid flow where there is an interface between two Ca = σ μ ⋅V (7) different fluids, especially for multiphase flows Where V − is the velocity scale. presenting curved surfaces, as it is the case of drop or If the velocity scale is given by the mass diffusion bubble migration with significant surface (D-isothermal mass diffusion coefficient), in a layer deformation. of depth d : Apart from the above dimensionless numbers V = D / d (8) there are others expressing also relations between Which gives: important physical actions involved in hydrodynamic σ ⋅d stability/instability of interfaces between fluids. These Ca = = Bo ⋅Ga (9) numbers could be obtained as composites of some μ ⋅ D already established dimensionless numbers. Galilei Another form using the same notation is the dimensionless number (Ga) is one of them and inverse form: results as the ratio of gravity forces divided by μ ⋅ D viscous forces[8], as ratio of dimensionless numbers Ca = = Bo Ga (10) σ ⋅ d Bo and Ga, or more simply forming a dimensionless group of quantities g, ν, D related to the same If the velocity scale is Marangoni velocityVMa , characteristic length scale d: the capillary number can be written in the form: 3 ()− ∂σ ∂C ⋅ β ⋅ d Bo g ⋅d Ca = (11) Ga = = (14) σ Ca ν ⋅ D In fluid dynamics, the Ca number in the form of In terms of characteristic time scales, the rel.(10) expresses the relative effect of viscous forces dimensionless number Ga could be written as it versus surface tension acting across an interface follows: between a liquid and a gas, or between two τ diff ⋅τ visc -5 Ga = (15) immiscible liquids. When Ca <10 it is possible to τ 2 neglect the effects of viscous forces compared to grav surface tension and this is specific to many liquids. Where: 2 The dimensionless number Ca is useful but needs τ diff = d D − is the time scale of mass again a specified value of the characteristic length d 2 which remains also to be established at an adequate diffusion; τ visc = d ν − is the time scale of magnitude, corresponding to necessities. viscosity; τ = (d g)1/ 2 − is the gravity time scale . The inversely written is convenient in treating the grav Ca and Ga in the same way in the stress balance at The Galilei number is used more frequently in boundaries. viscous flow and thermal expansion calculations, for Weber dimensionless number (We) [7] example to describe fluid film flow over walls. By characterizes the relative importance of the extension it could be used for the case of fluid flow at deformability of an interface, giving a measure of the interfaces on a side or the other side of interface. In relative importance of the inertia of fluids compared these cases the mass diffusion coefficient of the to its surface tension, according to the relation: considered solute and the kinematic viscosity of the considered fluid will be taken into account in Δρ ⋅V 2 ⋅ d We = (12) calculations. σ Where: ρ- is the density of the fluid; V- is the The gravity time scale considered τ grav fluid velocity; d-is the characteristic length; σ- is the represents in this case the time needed for a body to surface tension. In conditions of terrestrial gravity g , travel a distance d under the gravity acceleration g.

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Considering the Boussinesq approximation and In the solutal Marangoni number can be identified using the definitions of Rayleigh dimensionless a group presenting dimensions of velocity, called number and Galilei dimensionless number, a Marangoni velocity: parameter called Boussinesq parameter could be (− ∂σ ∂C) ⋅ ΔC ⋅ (d / L) V = (18) computed: Ma μ Ra Ga = α ⋅ ΔT (16) Where L is the horizontal, lateral extent of the From here it follows that liquid layer and d is the layer depth where there is a 1/ 2 τ grav τbuoy = ()α ⋅ ΔT difference of concentration ΔC .The ratio d/L represents the coefficient of slenderness of the liquid Further, other useful expressions of Ga number system and is used mostly in liquid bridges. If it is will be presented in the context of the present paper. considered d=L , which is acceptable for continuity

reasons and equal conditions of comparison, it results: 2.3. Dimensionless numbers derived (− ∂σ ∂C) ⋅ ΔC V = (19) from the tangential component of the free Ma μ slip boundary condition This is the case of many situations encountered in The Marangoni dimensionless number (Ma) is interfacial mass transfer of surface active solute linked to the Marangoni effect (sometimes also called soluble in immiscible liquid including the systems the Gibbs-Marangoni effect) which consists in the steel-slags. In order to specify this and to keep the mass transfer along an interface due to surface tension importance of dimension scale, if the solutal gradient. The surface tension gradient can be caused Marangoni effect is present on a depth d of the layer, by a concentration gradient or by a temperature a parameter can be written: gradient. The presence of a gradient in surface tension β = ΔC / d (20) will naturally cause the liquid to flow away from regions of low surface tension. The Marangoni Or generally, using cartezian coordinate z instead number for the solutal case, valid for isothermal of d, this became the slope of concentration profile: ∂C conditions, is the ratio of capillary force due to the β = (20’) surface tension gradient of solutal origin to the ∂z viscous drag in the flow. It can be defined in several The Marangoni dimensionless number reaches ways but an accepted form of larger utility is the the form: following [9]: (− ∂σ ∂C) ⋅ β ⋅d 2 ()− ∂σ ∂C ⋅ ΔC ⋅ L Ma = (21) Ma = (17) μ ⋅ D μ ⋅ D In this form the Marangoni number represents the Where: ∂σ/∂(C)-is the concentration coefficient of ratio between the capillary force due to the surface the surface tension or of the interfacial tension, related -1 -1 tension gradient and the viscous drag in the flow. to the surface active solute C, in N·m ·(mass%) or in It is interesting to show that noting: N·m-1·(mole fraction)-1; Δ(C)-is a characteristic ∂σ (C) ∂C concentration difference across the liquid layer of slag γ σ ()c = (22) or, after case, along its surface, expressed in terms of σ (C) solute (C) content, in mass% or mole fraction, or a The dimensionless solutal Marangoni number characteristic concentration of the liquid phase (C)0; L could be written using the form of Ca dimensionless - is a characteristic length, in m; ρ-is the density of the number from rel. (10 ) in the following form: phase, in kg/m3; D - is the mass diffusion coefficient −1 2 Ma = γσ ()c ⋅ ΔC ⋅Ca (23) of solute, m /s; μ=ρ·ν - is the dynamic viscosity of the considered phase, in Pa·s; ν-is the kinematic viscosity Using the form of Ca dimensionless number from of the considered phase, m2/s. rel. (9), the following form is obtained: Dealing with solutal surfactant problems the Ma = γσ ()c ⋅ ΔC ⋅Ca (24) corresponding Marangoni dimensionless number is This last relation remembers formally, in a certain also called elasticity number and is generally defined measure, to Ra = α ⋅ ΔT ⋅Ga from rel.(16) which accordingly for diffusion and adsorbtion processes. If refers to buoyancy effect. there are adsorbed surfactants, the product The characteristic time scale of the capillarity due ()∂σ ∂C C0 can be taken as a coefficient of elasticity to solutal gradient of surface tension is given by: in compression of the surfactant monolayer, where 1 2 ⎛ ρ ⋅d 3 ⎞ C denotes the excess surfactant surface concentration τ = ⎜ ⎟ (25) Ma ⎜ ∂σ (C) ∂C ⋅ΔC ⎟ and C0 is a reference value. ⎝ ⎠

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The time scale associated with capillary forces coefficient, interfacial instability versus stabilization acting at a deformable interface (Laplace force in the actions could also be well characterized by a new normal stress boundary condition) is the following: dimensionless number (Ni) introduced in the 1 2 following paragraphs in this paper for the case of ⎛ ρ ⋅ d 3 ⎞ τ = ⎜ ⎟ (26) complex effects of capillarity force at a unique length cap ⎜ ⎟ ⎝ σ ⎠ scale in depth and along the interface. It could be It results the ratio: derived combining the specific actions involved in the 2 Marangoni dimensionless number and in the Bond τ cap ∂σ (C) ∂C ⋅ ΔC = = γ ⋅ ΔC (27) dimensionless number (or Weber dimensionless 2 σ ()C number according to the case), rearranging the τ Ma σ (C) specific actions in order to put into evidence the ratio between destabilizing effect of capillarity force 3. Other possible groups proposed to (solutal, thermal or both two) due to the gradient of evaluate particularly the effects of the surface tension, in the form containing the capillarity actions at interface in presence concentration coefficient of the surface/interface of other physico-chemical and physical tension or the temperature coefficient of the actions surface/interface tension and the stabilizing effect of interface tension force. A dimensional number called Marangoni Unfortunately, the total difference from the other -2 dimensionless numbers, made impossible the coefficient NMa, in [m ] was introduced and argued by D. Agble and T.A.Mendes-Tatsis[10] based on a nomination of this new proposed dimensionless phenomenological approach as having the ability to number in other way than it was made in this paper. express, in a quantitative manner, the influence of Otherwise, it is known that nomination of factors that initiate or inhibit Marangoni convection, dimensionless numbers using the same basic symbols, when surfactant transfers from an aqueous phase A even when using sub-scripts, already introduces frequently strong confusions with negative effects in into an organic phase B. High values of NMa are understanding and handling in complex relations. obtained in conditions that promote Marangoni When it is written only for soluto-capillarity effect, convection and smaller values when conditions that the new proposed dimensionless number symbolized inhibit Marangoni convection are prevalent. (Ni) by the author of the present paper, reach the It presents a high numerical sensitivity and was form: proven as being able to predict stability in 29 cases Ni = Ma ⋅ Bo = −9 from 30 studied cases, when N > 10 m2. solutocapillarity force gravity force Ma = ⋅ = According to its definition it represents a ratio viscous forces int erface tension force between the surface tension and its associated effects solutocapillarity force gravity force which produce surface motion and the viscous and = ⋅ int erface tension force viscous forces related effects that restrict the surface movements: (29) dγ Ni = Ma ⋅ Bo = γ ⋅ ΔC ⋅Ga (30) ⎜⎛ ⎟⎞dγ solutal σ (C) γ N ∝ ⎝ ⎠ ⋅ Replacing the solutocapillarity effect by the Ma thermocapillarity effect an equivalent form could be μBDAB ()RMMsurfac tant RMM phase obtained. Csurfac tant Ni = Mathermal ⋅ Bo = γσ (T ) ⋅ ΔT ⋅ Ga (31) Γ In terms of the corresponding time scales of the (28) included physico-chemical actions, the dimensionless number Ni reaches the following form: There is a certain touch of tautology at the 2 τ cap ⋅τ visc ⋅τ numerator in the expression of the Marangoni Ni = diff (32) 2 2 coefficient NMa established according to [10], which τ Ma ⋅τ grav seems to be due mainly because of the non-linear The new dimensionless number could quantify dependence of the surface tension upon the better, the contributions of stabilizing actions versus concentration of the surface active and transferable destabilizing actions at interface between immiscible solute. liquids in conditions when the action of capillarity Starting from these aspects, but following the force (solutal or thermal) prevails over the other philosophy used in forming the Marangoni physico-chemical or pure physical actions.

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The new proposed dimensionless number Ni unusual compared to the requirement and benefits of addresses sharp interfaces, evaluating separately in the dimensionless analysis, it is difficult to be used in each phase the possibilities of instability to occur in other systems than those aqueous-organic, and it takes established conditions. This is possible if a convenient into account neither the whole panel of physico- length scale is established for all physico-chemical chemical actions present at interface, nor a whatever actions. Also, it allows a computation of the major characteristic length scale. factor relevant in industrial technology – the The new proposed dimensionless group characteristic concentration ( ΔC ) or temperature Ni = Ma ⋅ Bo and its different expressions showed in ( ΔT ) which could vary or could be modified by this paper, exhibits many and improved possibilities different internal or external processes or by different to evaluate the occurrence of Marangoni flow and intended actions. This dimensionless number, in the convection. mentioned conditions, leads to a more suggestive This new dimensionless number, in one of its comparison between the thermal and solutal computation forms contains the Galilei dimensionless capillarity effect because it contains simultaneously number which introduces a convenient characteristic both the interfacial tension and the concentration length scales allowing keeping the interface flat. Also coefficient of this quantity, or after the case the it contains the coefficient expressing the ratio between thermal coefficient. the product of the characteristic concentration and the This kind of treating specific problems based on concentration coefficient of interfacial tension and the new proposed dimensionless groups is not an isolated interfacial tension as in the dimensional Marangoni one especially when particular cases of flow must be coefficient. evaluated. As an example is the dimensionless number Bejan in fluid mechanics and heat transfer References representing the dimensionless pressure drop along a [1]. Nita P.S. - Materials Science and Engineering A, 495(2008) channel of length L and playing the same role in 320–325 forced convection as Rayleigh dimensionless number [2]. Nita P.S. - The Annals of “Dunarea de Jos” University of in natural convection [11]. Galati Fascicle IX Metallurgy and Materials Science 2, 2006, 112- 115. [3]. Ruzicka M.C. - Chemical Engineering Research and Design, 4. Conclusions 86(8) 2008, 835-868. [4]. Nita P.S. - The Annals of “Dunarea de Jos” University of Various dimensionless numbers derived from the Galati Fascicle IX Metallurgy and Materials Science 2 – 2009, normal component of the free slip boundary condition ISSN 1453 – 083X [4]. Nita P.S., Butnariu I, Constantin, N. - Revista de Metalurgia present a limited capability to express quantitative (Madrid), 46, 1(2010), 5-14. aspects at interface involved in hydrodynamic of [5]. Mahajan P.M.,Tsige M., Zhang S, Iwan D.A. Jr, Taylor interface with interfacial tension, because they are P.L., and Rosenblatt C. - Physical Review Letters 84(2) 2000, simple ratios of separate physical actions. 338-341. [6]. Scriven L.E., Sterling C.V. - Journal of fluid mechanics, Nevertheless they are able to give a characteristic 19(1964), 321-340. length scale and time scale if physical factors [7]. L.R. Weast, Astle D., Beyer W. - CRC Handbook of involved in their computations are well evaluated. Chemistry and Physics.(1989-1990), 70th ed. Boca Raton, Florida, Marangoni dimensionless number, despite useful CRC Press, Inc.. F-373,376. [8]. Wagner W. - Wärmeübertragung, 5th revised Edition; Vogel as proved until now in many cases, presents the major Fachbuch (1998) 119. difficulties in obtaining results of high degree of [9]. Molerkamp T. - Marangoni convection, mass transfer and validity and generality because it does not include the microgravity, doctoral thesis, ISBN 90-367-0982-2, Chapter 1 effects due to the presence of gravity as state and of Rijksuniversiteit Groningen, (1998), 6. [10]. Agble D., Mendes-Tatsis M.A. - International Journal of interfacial tension, both exerting stabilizing effects. Heat and Mass Transfer 44 (2001) 1439-1449. Proposal of the dimensional number under the [11]. S. Petrescu - International Journal of Heat Transfer and Mass form of Marangoni coefficient, besides the fact it is Transfer, Vol. 37, 1994, p. 1283.

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